Table Of ContentAN IMPROVED MULTIPLICITY CONJECTURE FOR CODIMENSION
THREE GORENSTEIN ALGEBRAS
JUAN C. MIGLIORE, UWENAGEL, ANDFABRIZIO ZANELLO
7
0
Abstract. The Multiplicity Conjecture is a deep problem relating the multiplicity (or degree)
0
of a Cohen-Macaulay standard graded algebra with certain extremal graded Betti numbers in its
2
minimal free resolution. In the case of level algebras of codimension three, Zanello has proposed a
n stronger conjecture. We provethis conjecture in the Gorenstein case.
a
J
1. Introduction
9
Thethird author has recently proposedan improvement of the so-called Multiplicity Conjecture,
]
C for the case of codimension three level algebras. The purpose of this note is to prove the first case
A of this improved conjecture, namely that of codimension three Gorenstein algebras. Furthermore,
. we characterize the cases when the improved conjecture for Gorenstein algebras is sharp.
h
t Let R be a polynomial ring over a field. Consider a Cohen-Macaulay standard graded algebra,
a
R/I, of codimension r with minimal free resolution
m
[ 0 → R(−j)βr,j → R(−j)βr−1,j → ··· → R(−j)β1,j → R → R/I → 0.
2 Mj≥0 Mj≥0 Mj≥0
v
Let M = max{j | β 6=0} and let m = min{j | β 6= 0}. The Multiplicity Conjecture of Herzog,
5 i i,j i i,j
8 Huneke and Srinivasan says that
4 r r
1 1
4
m ≤ e(R/I) ≤ M ,
0 r! i r! i
6 Yi=1 Yi=1
0 where e(R/I) is the multiplicity of R/I. It was shown by Huneke and Miller [8] that if the minimal
/
h free resolution is pure (i.e. for all i, β ·β = 0 whenever j 6= k) then both bounds above are
i,j i,k
at true (and thus are sharp). This conjecture has been shown to hold in a number of important
m cases, and it has been extended beyond the Cohen-Macaulay situation. We refer to the expository
: paper [4] for a more detailed (but at this point no longer up-to-date) history of the progress on
v
the problem. However, of greatest interest for us here is the following result of the first and second
i
X
author together with T. Ro¨mer:
r
a Theorem 1.1 ([9], Theorem 1.4). Let R/I be a standard graded Gorenstein algebra of codimension
three. Then the following lower and upper bounds hold:
(1) e(R/I) ≥ 1m m m + 1(M −M )2(M −m +M −m );
6 1 2 3 6 3 2 2 2 1 1
(2) e(R/I) ≤ 1M M M − 1 M (M −m +M −m ).
6 1 2 3 12 3 2 2 1 1
As an immediate consequence we get not only the fact the Multiplicity Conjecture holds for
codimension three Gorenstein ideals, but in fact that either bound of the Multiplicity Conjecture
is sharp for codimension three Gorenstein ideals if and only if the resolution is pure:
Corollary 1.2 ([9], Corollary 1.5). Let R/I be a graded Gorenstein algebra of codimension three.
Then the following conditions are equivalent:
(1) e(R/I) = 1m m m ;
6 1 2 3
(2) e(R/I) = 1M M M ;
6 1 2 3
1
2 J.Migliore,U.Nagel,F.Zanello
(3) R/I has a pure minimal free resolution.
Because of this result, it was conjectured in [10] and in [7] that in all cases, the bounds of the
Multiplicity Conjecture are sharp if and only if the resolution is pure. Special cases were proven in
[10], [7] and also in [11].
Corollary 1.2 of course gives a strengthening of the original conjecture. Other variants have
been suggested as well. One such idea was carried out by Francisco [3] and then from a different
point of view by the third author [12]. We give a brief outline of this approach. Suppose that R/I
is a Cohen-Macaulay standard graded algebra, and consider a minimal free resolution of R/I as
above. Ifthereis aredundantterminthisminimalfreeresolution (i.e. acopy ofR(−j)occurringin
consecutive free modules), then formally removing this term from both modules gives a new “Betti
sequence” that may or may not actually occur as a resolution for some Cohen-Macaulay algebra
R/I′. Nevertheless, one can formally compute the multiplicity as if this sequence were exact, and
it will equal the multiplicity of the original algebra R/I. However, the m potentially increase, and
i
the M potentially decrease. If the multiplicity formally computed with the new “Betti sequence”
i
satisfies the bounds with the new Betti numbers, then e(R/I) also satisfies the bounds of the
Multiplicity Conjecture. Thus the strengthened conjecture is that obtained by replacing the m
i
and the M by these new Betti numbers.
i
One drawback to this approach, in general, is that there may well be choices in the cancellation
of terms. For instance, there may be three consecutive copies of the same R(−j), so choosing
one pair gives a different result than choosing the other pair. Francisco’s approach is to make the
convention that the rightmost cancellation takes priority.
Zanello’s approach differs in two ways. First, herestricts to the case of levelalgebras of codimen-
sion 3. Recall that a level algebra of codimension 3 is a Cohen-Macaulay ring for which β = 0
3,j
for all but one value of j. With this restriction, there is no choice in the cancellation. Second, he
notes that in this case the strengthened conjecture can be phrased entirely in terms of the Hilbert
function of R/I. We give thedetails, notation and precisestatement of thestrengthened conjecture
in the nextsection. However, we would like to pointout that in computing the“Betti numbers”for
the improved bound, we may well be looking at a “Betti diagram” that does not exist. We will see
that this can happen even in the Gorenstein case. It was shown by Francisco that in his approach,
the reduction to diagrams that do not actually exist can producenumerical counterexamples to the
Multiplicity Conjecture. That this does not happen in the Gorenstein case (as we show) or more
generally in the level case (conjecturally) is remarkable.
The main result of this paper (Theorem 3.1) is that if R/I is a codimension 3 Gorenstein graded
algebra then it satisfies the strengthened conjecture. We use results of Diesel as well as the fact
that both bounds of the original Multiplicity Conjecture are known to hold ([6] Theorem 2.4,
[9] Theorem 1.4). Furthermore, we characterize those Gorenstein graded algebras for which the
strengthened conjecture is sharp.
2. Hilbert functions and cancelling redundant terms
In the recent paper [12], the third author sought to find new conjectural bounds which are
strongerthanthoseoftheMultiplicity Conjecture(ingeneral), andwhichcouldbeexpressedpurely
in terms of the Hilbert function of R/I. Because of the nature of this approach, the conjecture
is stated only for codimension three level algebras. However, as we will see, a similar idea was
proposed by Francisco also in higher codimension.
Let A= R/I be a codimension three Artinian algebra with Hilbert function
h = (1,3,h ,...,h ,h ).
2 c−1 c
Animprovedmultiplicityconjecture 3
It is a standard fact (see [5]) that the third difference of h has the following connection with the
graded Betti numbers of A:
∆3h(t) := h −3h +3h −h = β −β −β
t t−1 t−2 t−3 2,t 1,t 3,t
for all t ≥ 1. Notice that under most circumstances, this formula does not guarantee that we know
the values of any of the Betti numbers. This formula does not pick up the existence of “ghost” or
redundant terms in the minimal free resolution (i.e. terms R(−t) that are repeated in the second
and either the first or the third free module (or both) in the resolution).
However, we can draw the following conclusions:
(1) if ∆3h(t)> 0 then β > 0.
2,t
(2) The initial degree, m , of I can be read from the Hilbert function – it is
1
i+2
m = min i | h < .
1 i
(cid:26) (cid:18) 2 (cid:19)(cid:27)
In this degree of course β = β = 0, so ∆3h < 0. In fact m = min{t | ∆3h(t) < 0} and
2,t 3,t 1
β = −∆3h(m ).
1,m1 1
(3) Since A is Artinian, we have
−∆3h(c+3) = β = h and ∆3h(t) = 0 for all t > c+3.
3,c+3 c
If we further assume that A is level, then β = 0 for all t except t = c+3. Hence, in this
3,t
case, for all t < c+3 we have
∆3h(t) < 0 ⇒ β > 0.
1,t
Definition 2.1. Assume that A is level. We set
• n := min{t | ∆3h(t) < 0}.
1
• n := min{t > 0 | ∆3h(t)> 0}.
2
• N := max{t ≤ c+1 | ∆3h(t) < 0}.
1
• N := max{t | ∆3h(t) > 0}.
2
Remark 2.2. (1) Whether or not A is level, we have n = m .
1 1
(2) Since A is level, we have N =M .
2 2
(3) We also have n ≥ m and N ≤ M . What might prevent equality is the existence of
2 2 1 1
redundant terms.
(4) If A is level then, in the Multiplicity Conjecture, we have m = M = c+3, which can be
3 3
read uniquely from the Hilbert function.
Because of Remark 2.2, the third author made the following conjecture:
Conjecture 2.3 ([12], Conjecture 3.1). If R/I is level of codimension three then
1 1
n n (c+3) ≤ e(R/I) ≤ N N (c+3).
1 2 1 2
3! 3!
Note that this is a strengthening of the Multiplicity Conjecture. As pointed out in the introduc-
tion, the idea is similar to one described by C. Francisco in [3].
3. A proof of the conjecture for codimension three Gorenstein algebras
In this section we prove the main result of this paper:
Theorem 3.1. Conjecture 2.3 holds for codimension three Gorenstein algebras.
4 J.Migliore,U.Nagel,F.Zanello
Proof. We first need some notation and background results.
Using the structure theorem of Buchsbaum and Eisenbud [1], Diesel [2] gave a complete de-
scription of the degree matrices that can occur in the middle of the minimal free resolution of a
codimension three Gorenstein algebra. (She gave a number of consequences of this result, which
do not concern us here.) We set our notation, mostly following [2].
Let A= R/I be a graded, codimension 3 Artinian Gorenstein algebra over R := k[x,y,z], where
k is a field. From [1] we know that a minimal generating set for I has an odd number of elements.
So suppose that R/I has a minimal free resolution
n n
B
0→ R(−s)→ R(−p ) −→ R(−q ) → R → R/I → 0
i i
Mi=1 Mi=1
where n is odd. Buchsbaum and Eisenbud showed that B can be chosen to be skew-symmetric,
and the minimal generators are the pfaffians of B. The socle of R/I occurs in degree s−3, the q
i
are the degrees of the minimal generators of I and the p are the degrees of the minimal relations
i
among the generators. We make the convention that
q ≤ q ≤ ··· ≤ q
1 2 n
p ≥ p ≥ ··· ≥ p .
1 2 n
Notice that p +q = s for all i= 1,...,n. We consider the degree matrix
i i
p −q p −q ... p −q
1 1 2 1 n 1
p1−q2 p2−q2 ... pn−p2
p −q p −q ... p −q
δB = 1 3 2 3 n 3
.. .. ..
. . .
p1−qn p2−qn ... pn−qn
The entries of δB are increasing as one moves up and to the left.
The key fact for us is summarized by Diesel in [2], first paragraph of Section 2.3. We paraphrase
her observations as follows:
Under our hypotheses and notation, if δB contains a non-diagonal entry that is zero
then there is a second such entry. In this case we can allow these entries to be non-
zero constants c and −c. The resulting ideal of pfaffians will still be Gorenstein, of
height 3, and minimally generated by n−2 elements.
This says that whenever p = q for i6= j then the corresponding (numerically redundant) terms
i j
in the resolution have associated to them a second “dual” pair of numerically redundant terms,
and these four summands can be removed, with the strong conclusion that the result is still the
minimal free resolution of a Gorenstein algebra of codimension three.
We will consider the lower bound of Conjecture 2.3. The upper bound is proved in a completely
analogous fashion.
Given the graded Betti numbers of R/I, the computation of n is performed as follows. Start
2
with p (the smallest syzygy degree). If there is any q equal to p , with i 6= n, then we will also
n i n
have p = q . Thus the summands in the minimal free resolution corresponding to q ,p ,p ,q
i n i n i n
can be removed, and what remains is still the set of graded Betti numbers of a codimension 3
Gorenstein algebra A′ = R/I′. If no such q exists then n = p . Continuing in this manner, we
i 2 n
can end in one of two ways:
(1) at some point we have computed n by the nonexistence of q , or
2 i
(2) wecome toapointwheretheonly possiblecancellation comes fromq (suitably reindexed).
n
Animprovedmultiplicityconjecture 5
In the first case, note that n = m (A′) for this new Gorenstein algebra A′ obtained by formal
2 2
cancellation. But we know that the Multiplicity Conjecture holds for A′, by Theorem 1.1. So
1n n (c+3) = 1m m (A′)m
3! 1 2 3! 1 2 3
≤ e(A′)
= e(A).
Hence Conjecture 2.3 holds for A= R/I.
Now suppose that (2) holds above. So the graded Betti numbers allow some numerical cancella-
tion that does not occur for any Gorenstein algebra. However, diagonal entries in δB correspond
to pairs p ,q , while under our assumption we are always working with the syzygy of least degree.
i i
Hence we have for the last Gorenstein algebra constructed, that p = q ; that is, the syzygy of
n n
least degree coincides with the generator of largest degree. Furthermore, clearly we have reduced
the problem to a situation where β = β = 1 in order for no non-diagonal cancellations to be
2,pn 1,qn
possible in these degrees. We conclude that the Betti diagram must have the form
1 n n 1
1 - - -
. .
. .
. .
- a - -
1
... ...
- a - -
r−1
- a 1 -
r
- 1 a -
r
- - a -
r−1
. .
. .
. .
- - a -
1
. .
. .
- . .
- - - 1
This Betti diagram exists (thanks to Diesel), but the computation of n requires one more
2
cancellation, even though such an algebra does not exist (since it has to be Gorenstein, but has
an even number of minimal generators). However, note that the Betti diagram above is quasi-
pure, and the diagram obtained by removing the 1’s is even pure. It was shown by Herzog and
Srinivasan for Cohen-Macaulay algebras with quasi-pure resolution ([6], Theorem 1.2), and by the
first two authors and Ro¨mer for Cohen-Macaulay modules of rank 0 with quasi-pure resolution
([10], Theorem 4.2), that the Multiplicity Conjecture holds in those cases. Furthermore, as noted
by Francisco ([3], immediately after Theorem 4.7), the proof of this result “is numerical: Any
potential quasi-pure resolution below that of a lexicographic ideal satisfies the bounds; there is no
need for there to exist a module with that resolution.”
Weconcludethatthelower boundofConjecture2.3holds. Asremarkedearlier, theupperbound
is proved similarly, and in fact the steps are dual to those given for the lower bound (we start with
the generator of largest degree). (cid:3)
As one might expect, Conjecture 2.3 can be sharp even for Gorenstein algebras that do not
have pure resolution (in contrast to the situation of Corollary 1.2). It depends only on the Hilbert
function, and on whether the diagram obtained after performingall possible numerical cancellation
is pure. We omit the proof, which is immediate from the proof of Theorem 3.1; the only additional
ingredient is Corollary 1.2.
Corollary 3.2. For a Gorenstein algebra of codimension three, the following are equivalent:
(1) e(R/I) = 1n n (c+3).
3! 1 2
6 J.Migliore,U.Nagel,F.Zanello
(2) e(R/I) = 1N N (c+3).
3! 1 2
(3) n = N and n = N . (That is, the Betti diagram reduces to a pure one via formal
1 1 2 2
cancellation.)
(4) For all 1 ≤ t ≤ c + 2, there is exactly one t and one t for which ∆3h(t ) < 0 and
1 2 1
∆3h(t ) > 0 (all other ∆h(t)= 0); furthermore, −∆3h(t ) =∆3h(t ) and t +t = c+3.
2 1 2 1 2
Example 3.3. As remarked, the condition in Corollary 3.2 does not quite say that the minimal
free resolution of the Gorenstein algebra has to be pure. For instance, the Hilbert function
1, 3, 6, 6, 3, 1
leads to a third difference
1, 0, 0, −4, 0, 4, 0, 0, −1,
from which we immediately see that n = N = 3 and n = N = 5, and Conjecture 2.3 is sharp.
1 1 2 2
However, the general minimal free resolution for such a Hilbert function is
R(−5)4 R(−4)
0 → R(−8) → ⊕ → ⊕ → R → R/I → 0
R(−4) R(−3)4
which is clearly not pure.
Remark 3.4. One can check that the example above illustrates that there are situations when
bothboundsof Conjecture2.3 aresharpwhileneitherboundof Theorem1.1 issharp. Ontheother
hand, any Gorenstein graded algebra for which the minimalfree resolution is not pureand does not
have redundantterms clearly gives an instance whereTheorem 1.1 is sharper than Conjecture 2.3.
Acknowledgements. The research contained in this paper was performed during the third au-
thor’s visit to the first author at the University of Notre Dame, and that visit was supported by a
grant of the Vetensk˚apsradet (Swedish Research Council) and a grant of the Department of Math-
ematics of the University of Notre Dame. Moreover, the third author was funded by the Go¨ran
Gustafsson Foundation.
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Animprovedmultiplicityconjecture 7
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Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556
DepartmentofMathematics,UniversityofKentucky,715PattersonOfficeTower, Lexington,KY
40506-0027
Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden
